Usually we have axiomatic theory and the we look for model for it - this is book picture. Of course in real math usual one has a "model" that is given structure and looks for proper axiomatizing of it. SO it is interesting question:

Some structure called "model" is given by some (countable) number of axioms.

How many other axiomatic theories it may to be a model?

Are there any different theories ( non-isomorphic) from the first one?

Are this all theories related?

Some intuition: In this picture model is not "example of axiomatic structure" but "point of intersection of many axiomatic structures". How big is space in which this theories may cross?

3 Answers
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You are using the terms "model" and "theory" in an idiosyncractic way.

In model theory, a model is a first-order structure, that is, a set with some functions, relations and perhaps distinguished elements, called constants. A theory, in contrast, is a collection of assertions, a set of sentences in this language. A given theory, which can be thought of as a set of axioms in the sense that you mentioned, can give rise to many models. And indeed, the Lowenheim-Skolem theorem says that if a theory has an infinite model, then it has infinite models of arbitrarily large cardinality. (Thus, except in trivial cases one cannot uniquely specify a model by giving "some (countable) number of axioms" as you said, since the same axioms will have models of many different sizes.)

Suppose that M is a model in a language of size κ, meaning that the language has κ many possible assertions. In this case, since any given assertion is either true in M or its negation is true in M, the complete theory of M, that is, the set Th(M) consisting of all sentences true in M, will also have size κ. Any subset S of Th(M) will also be true in M, of course. Thus, there are 2κ many theories true in M. For example, if the language has countably many symbols in it, then any given model in this language will satisfy continuum 2ω many theories.

But this answer counts theories as different, when they are different merely as sets of sentences, even when these theories have the same models. But for the purposes of counting theories, it may be more sensible to use another common definition of theory, which is a set of sentences closed under consequence. This amounts to identifying theories that have the same models.

With this second understanding of theory, the answer is a little more subtle. In the empty language, for example, every model is just a naked set, with no structure. There are exactly countably many countable models in this language: one of each finite size and one countably infinite model. If φn is the assertion that there are exactly n objects, then for any set A of natural numbers, we may form the theory TA, which asserts that ¬φn for each n in A. These theories are all inequivalent, and all true in any infinite model. If M is any model, then there are continuum many theories TA that are true in M.

This shows that in fact every model M, in any language, satisfies at least continuum many deductively closed theories.

If the language is larger, with uncountable size κ, then either there are uncountably many relation symbols, uncountably many function symbols or uncountably many constant symbols. In each case, it is a fun exercise to form 2κ many inequivalent theories T in the language. Given any model M, let σ be any sentence false in M. For any theory T containing σ, we may form the theory T' = { σ implies φ | φ in T }. This theory is true in M, since σ is false in M. Thus, by counting theories in this manner, one can show that there are 2κ many inequivalent theories true in M.

"This shows that in fact every model M, in any language, satisfies at least continuum many deductively closed theories. " So we are completely lost in the space of models. Are there some minimal closed theories which are true in given model? Or all of them are completely different?
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kakazFeb 8 '10 at 15:09

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The theories T_A are different, in the sense that there are models distinguishing any two of them, but they are all completely trivial. They just say that the structure has a size that is not in A. The theories T' in my last paragraph, however, are not trivial. These theories assert what the model could have been like, if sigma had been true. Since sigma is not true, these theories are vacuously true in M, but the theories are really talking about what might have been true in another structure. I take this to show that we should investigate all models, rather than theories true in one model.
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Joel David HamkinsFeb 8 '10 at 15:26

(Note that you first need to fix a first-order language L for your structures, otherwise you can't talk about models and theories. It is not necessary for that language to be countable.)

Since every sentence of the language L is either true or false in your structure M, there is only one complete theory that M satisfies, namely the theory Th(M) of all sentences of the language which are true in M. However, every subset of Th(M) is also a theory that M satisfies; those are precisely all the theories that M satisfies.

Interesting: but it is known if this theories may be grouped in some way? It is obvious that given model ( even with one element) is not a model for all possible theories: so there are theories for which it is not a model;-). So assuming we have infinitely many theories, are there some, I do not know, invariants, orders, lattice operations? which allow as to show whether there is some structure in such big space of theories? Maybe from Category theory point of view it is somehow interesting? Or it is just another trivial question?
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kakazFeb 8 '10 at 12:57

And Francois pointed out you don't even need the binary relation!
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Joel David HamkinsFeb 8 '10 at 19:27